What is the Riemann Hypothesis REALLY about?

Solve one equation and earn a million dollars! We will explorer the secrets behind the Riemann Hypothesis - the most famous open problem in mathematics - and what it would tell us about prime numbers.

I should have mentioned one additional property, namely zeros are mirrored along the line 1/2, even though non of them are found and maybe even non of them even exist. This way, every zero not on the line would giva a harmonic with Re(s) greater than 1/2, thereby breaking the estimates for the prime counting function.

Results discussed / references
Consequences of different zero-free regions on the growth of |pi(x)-li(x)|: A.E. Ingham: The Distribution of Prime Numbers, Cambridge University Press
Scene from Big Bang Theory: S12E6 The Imitation Perturbation

[u.v.s.]

One mistake in this video: It is not true that Riemann knew the first 3 roots and made the hypothesis. He had over 600 zeros of this type, because he used what is known to us all as the Riemann-Siegel formula rediscovered really long after Riemann's death (the zeros are really difficult to compute without a computer, and 80 years after death of Riemann the world of mathematics knew less than 100 first nontrivial zeros. It was quite a shock to discover that Riemann himself had calculated many more than the rest of the world)

[Pjx]

“Ok guys, if this is the first time you heard of these ‘imaginary numbers’, let’s talk of a simple topic involving them: the Riemann Zeta Function”

That was a hell of a leap!

[paradoxicallyexcellent]

Holy. Shit. This video is CRIMINALLY underwatched.

Sharing it far and wide. I am a math phd (now in a different field) and, although I studied analysis, it is astounding that no one ever could explain to me, as well as you just did, how the Riemann Hypothesis actually matters to the study of prime numbers. Years of casual lectures and conversations. No one approached the explanation with your clarity. I have absolutely crazy respect for your ability to communicate this. Just. Wow.

[nkdibai]

My mind exploded when you showed how the Riemann Conversion of the subtraction of the pole in s = 1 and the non-trivial zeros of the Riemann Zeta Function approached the distribution function of primes 🤯

[johnchessant]

For those wondering, the zeta function has a reflection formula such that the zeros in the critical strip have reflection symmetry across the critical line. i.e. say if s = 0.49 + 100i is a zero, then so is s = 0.51 + 100i. And it's that zero with the real part greater than 1/2 that would mess up that x^(1/2) error bound.

[johnchessant]

Excellent video! I always found it kinda frustrating for math popularization that _the_ million-dollar question was not only so hard to explain (see the 3b1b video) but also even harder to understand why mathematicians care (I mean, you basically need an entire semester of analytic number theory to go through all the details of this connection between Riemann zeta and primes). Kudos to you for being able to boil it down with some incredible animations!

[whitestonejazz]

I've been looking for something like this for a while. I always wanted a Riemann hypothesis video that went a bit more deeply into the math. The concrete examples were really helpful too; like doing the error calculation for pi(10^50) or showing the sum of the first 200 harmonics. Great stuff

[hippyhair]

Thank you for making this. This is the only video I have seen that actually explains HOW the Zeta function actually contributes to primes in how it is constructed. All the other have just said "this allows you to know more about primes", but this was really clear and informative, thank you.

[meliniak]

This is the best video on Riemann hypothesis I've seen on YT. Congratulations on explaining it in-depth yet in simple terms.

[Axacqk]

Explaining the basics of complex numbers and RH in one video. Man, you're a brave soul.

[InfiniteRegress]

Thank you for another excellent video! ^_^

If I may attempt to add something of value, Robin's Theorem is a relatively easy to understand statement about an inequality whose truth for all positive integers greater than 7 factorial (5040) is equivalent to the truth of the Riemann Hypothesis.

That is, sigma(n) < n * ln(ln(n)) * e^gamma

The inequality holds that the sum of the divisors of an integer, n, is less than the product of n with the natural logarithm of the natural logarithm of n, as well as e raised to the Euler-Mascheroni constant "gamma", with the 27 exceptions of 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360, 720, 840, 2520, or 5040.

Robin's Theorem states that the Riemann Hypothesis is true if 5040 is the final exception, and is false if there are any more.

So, for those really interested in that $1, 000, 000, here is another way to approach it.

[F.E.Terman]

I've seen many videos on this, and quite enjoyable they were too, but this is the first one that explains how all the bits fit together. Thank you!

[exponentmantissa]

Excellent. First video of yours I have seen. I am a retired electrical engineer that hung out with physicists my whole life. I find mathematics fascinating, luckily I have good math skills and I thought I knew a fair bit about the Riemann function and hypothesis. The stuff here is refreshing and new to me. Great job!!!

[MattMcIrvin]

If you instead add simple periodic waves for all the nontrivial zeroes (a Fourier transform), you get a result that has sharp spikes at all the POWERS of primes, that is, all p^n where p is prime and n is a positive integer (plus a continuous component). It is very odd.

[rfvtgbzhn]

7:54 I am a physicist and I don't think that the term "imaginary numbers" is misleading. Physicists use them a lot, but just because they make some computations easier. But they are never measured, unlike real numbers. Also even in quantum mechanics, where they appear in the wave function, one could use R² instead C to get rid of them as there is an isomorphism between R² and C. This isomorphism is used for example for all the diagrams of complex numbers used in this video (by identifying the real numbers with the x-axis and the imaginary numbers with the y-axis).

[KStarGamer_]

4:25 is a bit misleading. This standard Mellin transform representation of the Riemann zeta function only converges for Re(s) > 1, like the standard series expression for zeta. Thus, saying that we are trying to find the roots of this representation is misleading since the zeros of zeta are all behind Re(s) = 1 in real part.

[kapoioBCS]

Without the knowledge from ring theory, people will never understand the true deepness of primeness as a general notion.

[quackcharge]

I'm very moronic when it comes to math but this was a joy to watch. I didn't fully grasp 5% of what you showed here but it made me want to understand more, seeing the prime frequencies emerge from the subtraction was absolutely beautiful, thank you!

[ericvosselmans]

Riemann must have been an incredibly brilliant and intelligent man, to have seen all this with only 3 zeroes and no computers to work with. What an amazing genius

[drewj-ui]

After watching at least 5 videos, I finally have a better understanding of the connection of the zeta func. to the prime numbers, thank you!